Solve ∫ (from 0 to 3) of (2+0.33x)(0.67x) wrt x

$$\int_{ 0 }^{ 3 } (2+0.33x)(0.67x) d x$$

Evaluate

$$8.0199$$

Solution Steps

Use the distributive property to multiply 2+0.33x by 0.67.

$$\int _{0}^{3}\left(1.34+0.2211x\right)x\mathrm{d}x$$

Use the distributive property to multiply 1.34+0.2211x by x.

$$\int _{0}^{3}1.34x+0.2211x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{67x}{50}+\frac{2211x^{2}}{10000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{67x}{50}\mathrm{d}x+\int \frac{2211x^{2}}{10000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{67\int x\mathrm{d}x}{50}+\frac{2211\int x^{2}\mathrm{d}x}{10000}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 1.34 times \frac{x^{2}}{2}.

$$\frac{67x^{2}}{100}+\frac{2211\int x^{2}\mathrm{d}x}{10000}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 0.2211 times \frac{x^{3}}{3}.

$$\frac{67x^{2}}{100}+\frac{737x^{3}}{10000}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{67}{100}\times 3^{2}+\frac{737}{10000}\times 3^{3}-\left(\frac{67}{100}\times 0^{2}+\frac{737}{10000}\times 0^{3}\right)$$

Simplify.

$$\frac{80199}{10000}$$